Spectral Distances: Results for Moyal Plane and Noncommutative Torus

The spectral distance for noncommutative Moyal planes is considered in the framework of a non compact spectral triple recently proposed as a possible noncommutative analog of non compact Riemannian spin manifold. An explicit formula for the distance between any two elements of a particular class of pure states can be determined. The corresponding result is discussed. The existence of some pure states at infinite distance signals that the topology of the spectral distance on the space of states is not the weak $*$ topology. The case of the noncommutative torus is also considered and a formula for the spectral distance between some states is also obtained.